A clinical program is a program that allows law students to represent actual clients (typically indigent and/or criminal clients), under close supervision by attorneys. Students do everything attorneys do (except bill for their time), including drafting pleadings, negotiating, and court appearances.
Why do it? Most law school classes are geared toward teaching you broad legal concepts, and how to “think like a lawyer.” That's important stuff to know, but it also helps to learn some practical information and skills. Clinical programs teach you How To Be A Lawyer – how to deal with clients, how to deal with opposing attorneys, how to prepare for and conduct a trial, how to resolve ethical issues, etc. Also, if you are in the jurisdiction where you ultimately plan to practice, you learn local process – the steps in a divorce case, eviction procedures, etc.